Chapter 4

Task 1-3

data("Boston")
# explore the dataset
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim(Boston)
## [1] 506  14
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
# calculate the correlation matrix and round it
cor_matrix<-cor(Boston) %>% round(digits = 2)

# print the correlation matrix
cor_matrix
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00
# visualize the correlation matrix
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex = 0.6)

Boston Dataset describes Housing Values in Suburbs of Boston. Here positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients.

f.ex. Age is strongly negatively correlated with weighted mean of distances to five Boston employment centres (dis). And index of accessibility to radial highways (rad) is strongly positively correlated with property tax. Number of rooms is positively corrrelated with median value of homes, whereas lower status of the population is strongly negatively correlated.

Task 4

# center and standardize variables
boston_scaled <- scale(Boston)

# summaries of the scaled variables
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
# class of the boston_scaled object
class(boston_scaled)
## [1] "matrix"
# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)

# summary of the scaled crime rate
summary(boston_scaled$crim)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.419367 -0.410563 -0.390280  0.000000  0.007389  9.924110
# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))

# look at the table of the new factor crime
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)

# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)

# number of rows in the Boston dataset 
n <- nrow(boston_scaled)

# choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# create train set
train <- boston_scaled[ind,]

# create test set 
test <- boston_scaled[-ind,]

# save the correct classes from test data
correct_classes <- test$crime

# remove the crime variable from test data
test <- dplyr::select(test, -crime)

In the scaling we subtract the column means from the corresponding columns and divide the difference with standard deviation so the values become z-scores. This normalizes the data so now it will be normally distributed. For some multivariate techniques such as multidimensional scaling and cluster analysis, the concept of distance between the units in the data is often of considerable interest and importance. When the variables in a multivariate data set are on different scales, it makes more sense to calculate the distances after some form of standardization.

Task 5

# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)

# print the lda.fit object
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2400990 0.2623762 0.2574257 0.2400990 
## 
## Group means:
##                  zn      indus        chas        nox           rm
## low       0.9104661 -0.9013355 -0.06938576 -0.8765661  0.359531557
## med_low  -0.1148968 -0.3096663 -0.01233188 -0.5582234 -0.177882713
## med_high -0.3783982  0.2305261  0.21980846  0.4410913  0.009210396
## high     -0.4872402  1.0149946  0.01179157  1.0676544 -0.444707523
##                 age        dis        rad        tax     ptratio
## low      -0.8821896  0.8461996 -0.6941624 -0.7399639 -0.43041372
## med_low  -0.2740952  0.3618551 -0.5365694 -0.4880028 -0.08012124
## med_high  0.3873672 -0.3852754 -0.4142635 -0.3015008 -0.39339902
## high      0.7895559 -0.8564030  1.6596029  1.5294129  0.80577843
##                black       lstat       medv
## low       0.38461631 -0.74343843  0.4928914
## med_low   0.32772645 -0.09690793 -0.0374169
## med_high  0.06109754  0.01463829  0.1377818
## high     -0.75852677  0.83891606 -0.6272506
## 
## Coefficients of linear discriminants:
##                  LD1          LD2          LD3
## zn       0.156800031  0.724984869 -0.960076683
## indus    0.043970715 -0.366217879  0.190385495
## chas    -0.002181548  0.004329863  0.066021868
## nox      0.291493125 -0.680999256 -1.357597624
## rm       0.035147078 -0.075810844 -0.156924102
## age      0.246672909 -0.289748569  0.161791516
## dis     -0.164191602 -0.250953678  0.316276875
## rad      3.431851363  0.776793588  0.003745442
## tax     -0.032365423  0.134350552  0.510233711
## ptratio  0.185236478  0.132860643 -0.274414656
## black   -0.155237026  0.012244395  0.148647862
## lstat    0.184698460 -0.182224623  0.358585069
## medv     0.088383494 -0.277414853 -0.212355238
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9528 0.0351 0.0121
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(train$crime)

# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)

Here Linear Discriminant 1 explains most of the between group variance.

rad: index of accessibility to radial highways is the most significant factor that predicts higher crime rate.

Task 6

# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       20       8        2    0
##   med_low    6      10        4    0
##   med_high   0       8       13    1
##   high       0       0        1   29

This prediction is very good at predicting high crime rates, but worse at predicting med_low or med_high correctly. ’ ##Task 7

data("Boston")
new_boston <- scale(Boston)
# k-means clustering
km <-kmeans(new_boston, centers = 4)

# plot the Boston dataset with clusters
pairs(new_boston, col = km$cluster)

#K-means might produce different results every time, because it randomly assigns the initial cluster centers. The function set.seed() can be used to deal with that.
set.seed(123)

# determine the number of clusters
k_max <- 10

# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(new_boston, k)$tot.withinss})

# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')

# k-means clustering
km <-kmeans(new_boston, centers = 2)

# plot the Boston dataset with clusters
pairs(new_boston, col = km$cluster)

One way to check optimal amount of clusters is to look at the total within cluster sum of squares (twcss). When twcss drops a lot the optimal number of clusters is found. Here twcss drops around 2. In the LDA there were two “main clusters” one with high crime and some points from med_high and the other larger cluster was the rest of the data points. This k-means clustering on this data also works best with 2 clusters.

Bonuses

km <-kmeans(new_boston, centers = 3)

# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)

# print the lda.fit object
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2400990 0.2623762 0.2574257 0.2400990 
## 
## Group means:
##                  zn      indus        chas        nox           rm
## low       0.9104661 -0.9013355 -0.06938576 -0.8765661  0.359531557
## med_low  -0.1148968 -0.3096663 -0.01233188 -0.5582234 -0.177882713
## med_high -0.3783982  0.2305261  0.21980846  0.4410913  0.009210396
## high     -0.4872402  1.0149946  0.01179157  1.0676544 -0.444707523
##                 age        dis        rad        tax     ptratio
## low      -0.8821896  0.8461996 -0.6941624 -0.7399639 -0.43041372
## med_low  -0.2740952  0.3618551 -0.5365694 -0.4880028 -0.08012124
## med_high  0.3873672 -0.3852754 -0.4142635 -0.3015008 -0.39339902
## high      0.7895559 -0.8564030  1.6596029  1.5294129  0.80577843
##                black       lstat       medv
## low       0.38461631 -0.74343843  0.4928914
## med_low   0.32772645 -0.09690793 -0.0374169
## med_high  0.06109754  0.01463829  0.1377818
## high     -0.75852677  0.83891606 -0.6272506
## 
## Coefficients of linear discriminants:
##                  LD1          LD2          LD3
## zn       0.156800031  0.724984869 -0.960076683
## indus    0.043970715 -0.366217879  0.190385495
## chas    -0.002181548  0.004329863  0.066021868
## nox      0.291493125 -0.680999256 -1.357597624
## rm       0.035147078 -0.075810844 -0.156924102
## age      0.246672909 -0.289748569  0.161791516
## dis     -0.164191602 -0.250953678  0.316276875
## rad      3.431851363  0.776793588  0.003745442
## tax     -0.032365423  0.134350552  0.510233711
## ptratio  0.185236478  0.132860643 -0.274414656
## black   -0.155237026  0.012244395  0.148647862
## lstat    0.184698460 -0.182224623  0.358585069
## medv     0.088383494 -0.277414853 -0.212355238
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9528 0.0351 0.0121
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(km$cluster)

# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)

rad is the most influencial linear separator for the clusters. It looks like the k-means clustering with 4 clusters makes clusters where low, med_low, med_high, high etc. mix a lot so the clustering is not perfect.

library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
model_predictors <- dplyr::select(train, -crime)

# check the dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)

plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color= ~train$crime)
#plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color=~classes)

Can’t get the color with km$cluster to work…